book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 5 TABLE 1: Signal Sets for Some Digital Modulations Schemes NAME OF MODULATION SCHEME SIGNAL SET: 0 ≤ t ≤ T s Binary phase-shift keying (BPSK) 1 ⇒ A c cos ( 2π f 0 t ) ; f 0 = n/T s , T s = T b , n integer 0 ⇒−A c cos ( 2π f 0 t ) Binary differential phase-shift keying Binary bit stream differentially encoded (DE); (DPSK) DE bits BPSK modulate the carrier. DE: data-bit 1 encoded as no change from reference bit; data-bit 0 encoded as a change from reference bit; current encoded bit is reference for next encoded bit. Binary frequency-shift keying (BFSK) A c cos ( 2π f 0 t ) , A c cos [ 2π ( f 0 + f ) t ] ; f 0 = n/T b ,f = m/T b , m, n integers, m = n M-ary phase-shift keying (MPSK) A c cos ( 2π f 0 t +2 ( i −1 ) π/M ) , M = 4 called quadriphase-shift keying i = 1, 2, ,M (QPSK) f 0 = n/T s , n integer M-ary frequency-shift keying (MFSK) A c cos [ 2π ( f 0 + ( i −1 )  f ) t ] ; f 0 = n/T s ,f = m/2T s (m ≥ 1 for orthogonal signals) a communication system’s performance in terms of bit error probability versus signal-to-noise ratio, usually specified as E b /N 0 , where E b is the bit energy for the signal (E b = E s / log 2 M for an M-ary system, where E s is the symbol energy) and N 0 is the one-sided power spectral density of the white, Gaussian background noise at the receiver input. Table 2 summarizes the bandwidth and communications efficiencies in additive white Gaussian noise (AWGN) for various digital modulation schemes. In the preceding discussion, it was presumed that the channel imposes a fixed attenuation and the only signal impairment was the AWGN at the receiver input (modeled as entering the system at this point because that is where the signal is weakest). Another common channel model is the one with time varying attenuation, perhaps due to obstructions or reflections, of the signal. If these attenuation variations are slow enough, they can be viewed as fixed throughout a book Mobk087 August 3, 2007 13:15 6 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION TABLE 2: Bandwidth and Communication Efficiencies of Some Digital Modulation Schemes NAME OF BANDWIDTH BIT ERROR E b /N 0 REQUIRED MODULATION EFFICIENCY PROBABILITY, P b FOR SCHEME (BPS/HZ) P b = 10 −6 Binary phase-shift 0.5 Q   2E b /N 0  a E b /N 0 = 10.53 dB shift keying (BPSK) Binary 0.4 coherent, Q  √ E b /N 0  coherent, 13.54 dB coherent frequency-shift 0.25 0.5exp [ −E b / ( 2N 0 )] 14.2dBnoncoherent keying (BFSK) noncoherent noncoherent Binary 0.5 0.5exp ( −E b /N 0 ) 11.2 dB differential (DE bit stream, see Table 1; phase-shift 0 sent as π-‘rad phase shift; keying (DPSK) 1 sent as 0 rad phase shift) M-ary DPSK 0.5log 2 M ≈ 2 m  1 +cos ( π/M ) 2cos ( π/M ) ×Q  2m  1 −cos  π M  E b N 0  m = log 2 M, M > 2 11.2dB, M = 2 12.9dB, M = 4 16.8dB, M = 8 21.4dB, M = 16 26.3dB, M = 32 M-ary phase- 0.5log 2 M ≈ Q   2E b /N 0  ; M = 2, 4 < 2Q   2m  E b N 0  sin π M  m = log 2 M (bound tight for M > 4) 10.5dB, M = 2, 4 14 dB, M = 8 18.5dB, M = 16 23.4dB, M = 32 28.5dB, M = 64 shift keying (MPSK) M-ary 2log 2 M M +3 frequency-shift coherent keying (MFSK) log 2 M 2M M 2 Q   log 2 M  E b N 0   coherent, M 2 ( M −1 ) M−1  k=1 ( −1 ) k+1 k +1  M −1 k  ×exp  −k log 2 M k +1 E b N 0  13.5dB, M = 2 10.8dB, M = 4 9.3dB, M = 8  coherent 14.2dB, M = 2 11.4dB, M = 4 9.9dB, M = 8  noncoherent noncoherent noncoherent a Q ( x ) =  ∞ x exp ( −u 2 /2 ) √ 2π du =  π/2 0 exp  − u 2 2sin 2 φ  dφ π  exp ( −x 2 /2 ) x √ 2π , x > 4. book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 7 given signaling interval. Perhaps the most frequently used model is the slow flat Rayleigh fading model, wherein a given transmitted signal is attenuated by a fixed (for that symbol interval) level modeled as a Rayleigh-distributed random variable and the next transmitted signal is likewise attenuated by a new, independent Rayleigh random variable, etc. For sufficiently slow fading, this model can be a fairly accurate representation of the true state of affairs, and it is simple to analyze a digital transmission system experiencing such a channel. The analysis proceeds by using the error probability expressions of Table 2 and averaging over the signal- to-noise ratio, E b /N 0 , not with respect to a Rayleigh probability distribution, but with respect to an exponential probability distribution because E b /N 0 = A 2 c T b /2N 0 , where A c is the signal amplitude which is modeled as a Rayleigh random variable. Thus, E b , being proportional to the signal amplitude squared, is exponentially distributed. This results in a particularly simple integral to evaluate in the case of binary DPSK or NFSK. For the latter case, P b,NFSK =  ∞ 0 1 2 exp ( −z/2 ) 1 Z exp  −z/ Z  dz = 1 2 + Z , (2.2) where Z is the average received E b /N 0 . For DPSK, the integration is similar. For BPSK the integral is challenging but possible to perform. The results for these two cases are P b,DPSK = 1 2  1 + Z  ; P b,BPSK = 1 2   1 −  Z 1 + Z   . (2.3) The sobering fact about the effects of Rayleigh fading is the penalty imposed on com- munications efficiency. The difference between signal-to-noise ratios for fading and nonfading cases for a given modulation scheme is called the fading margin for that scheme. For a bit error probability of 10 −3 , the fading margins for binary NFSK, DPSK, and BPSK are 16.04 dB, 19.05 dB, and 20.19 dB, respectively. For MPSK with M = 8 and 16, the fading margins are 15 dB and 14.6 dB, respectively. The question of what do about the penalty imposed by fading has a partial answer in the use of diversity, that is, providing several alternative paths to pass the signal through, not all of which will fade simultaneously, hopefully. 3 TYPES OF SPREAD SPECTRUM MODULATION The two mostcommon types ofspread spectrum modulationare direct-sequence andfrequency- hop spread spectrum (FHSS). A binary direct-sequence spread spectrum (DSSS) scheme was used in the illustrations of Fig. 1. book Mobk087 August 3, 2007 13:15 8 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION () 1 ct × () 2 ct− × () s t × () 1 d ct T− × × () 2 d ctT−− × () 0IF 2cos t ωω φ  ++  () dt () d s tT− () dt 0 2cos P t ω () 0 cos d P tt ωθ  +  () 0 sin d P tt ωθ  +  () 0IF 2sin t ωω φ   ++   FIGURE 3: Block diagrams of the transmitter (a) and receiver (b) for QPSK spreading with arbitrary phase modulation [1]. 3.1 QPSK Spreading With Data Phase Modulation Modulation types other than BPSK may be used in DSSS communication systems, both for the data and for the spreading. For example, Fig. 3 shows a transmitter/receiver structure for QPSK spreading with arbitrary data phase modulation. 3.2 Frequency-Hop Spread Spectrum As its name implies, FHSS involves hopping the data-modulated carrier pseudo-randomly in frequency. A combination of direct sequence and frequency hop modulation is often referred to as hybrid spread spectrum modulation. Another type of spread spectrum modulation, called time-hopped or pulse-position-hopped [3], involves time hopping the transmitted data pulses pseudo-randomly in time with respect to a fixed reference position for each signaling interval. While not prevalently implemented in the past, this type of spread spectrum is more popular recently because of the current intense exploration of ultra-wideband modulation techniques. book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 9 × () s t () d s tT− () dt 0 2cos P t ω × () dt ∧ FIGURE 4: Block diagram of a FHSS transmitter (a) and receiver (b) [1]. The focus of attention in this section is on FHSS modulation since the idea of DSSS was explained in relation to Fig. 1. A schematic block diagram of a FHSS communication system is shown in Fig. 4. Often, a noncoherent data modulation scheme, such as noncoherent FSK or DPSK, is used since it is more difficult to build frequency synthesizers that maintain phase coherence from hop to hop than those that do not. A pseudo-random code generator is used as a driver for a frequency synthesizer at the transmitter to pseudo-randomly hop the carrier frequency of the data modulator output. In keeping with the basic idea of spread spectrum, the hopping frequency range is quite broad compared with the modulated data bandwidth. The time interval of a frequency hop is called the hop period, T h . Two situations can prevail: the hop period can be long with respect to a data bit period; the hop period can be short with respect to a data bit period. The former case is referred to as slow frequency hop, and the latter case is referred to as fast frequency hop. Perhaps the most common situation in practice is slow frequency hop. Fast frequency hop has some advantages over slow frequency hop but is more difficult to implement. At the receiver, a pseudo-random code generator identical to the one used at the transmitter is implemented and used to drive a frequency synthesizer like the one used at the transmitter. Assuming that the pseudo-random number sequence output by the number book Mobk087 August 3, 2007 13:15 10 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION generator can be synchronized with the one at the transmitter (accounting for channel delay), the frequency hopping sequence will track that of the transmitted hopping sequence and the re- ceived frequency-hopped spread spectrum signal will be de-hopped whereupon an appropriate data demodulator can be used to recover the data sequence. In the early days of spread spectrum, FHSS was used to realize wider spread bandwidths than possible with DSSS systems. If the features of FHSS and DSSS are combined, the result is referred to as hybrid spread spectrum. Usually, the additional implementation complexity does not warrant the hybrid approach, so the actual use of such systems is seen very little. One advantage of the hybrid approach is to force a potential hostile interceptor to use a more complex interception strategy [4]. Example 1. A binary data source emits binary data at a rate of R b = 10 kbps. Each bit is sent in a DSSS communication system either as a 127-chip code as is (data bit 1) or inverted (data bit 0). (a) What is the bandwidth of the DSSS transmitted signal? (b) Compare this with a FHSS system that uses binary NFSK modulation. How many frequency hop slots are required to provide roughly the same transmission bandwidth as the DSSS system? Solution: (a) From Table 2, the bandwidth efficiency of BPSK is 0.5 which means that the trans- mission bandwidth of the unspread signal is 10/0.5 = 20 kHz. The spread signal bandwidth is 127 times of this or 2.54 MHz. (b) From Table 2, the bandwidth efficiency of binary NFSK is 0.4 which gives a trans- mission bandwidth for the unspread signal of 10/0.4 = 25 kHz. The number of frequency hops needed to provide the same spread bandwidth as the DSSS system is 2, 540, 000/25, 000 = 101.6 which is rounded to 102. The spread bandwidth of the FHSS system is therefore 2.55 MHz, which is close to that of the DSSS system. 3.3 Summary The previous two sections have laid the ground work for the consideration of spread spectrum communication systems with discussions of the basic idea of a direct sequence spread spectrum system and some of its features, a review of basic digital modulation techniques and, in ad- dition to the DSSS system example, descriptions of two generic spread spectrum modulation techniques—QPSK spreading and frequency-hop spread spectrum. book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 11 4 SPREADING CODES An important component of any spread spectrum system is the pseudo-random spreading code. Many options exist for the generation ofsuch codes, onlyafew of whichwillbe described here.In particular, m-sequences will first be described in terms of their generation and properties. Then, Gold codes will be discussed in terms of their generation and cross-correlation properties. Next, the small set of Kasami sequences will be introduced, followed by an introduction to quaternary sequences. Finally, the construction of Walsh functions will be illustrated. 4.1 Generation and Properties of m-Sequences Maximal-length sequences, or m-sequences, are simple to generate with linear feedback shift- register circuits and have many nice properties. But, they are relatively easy to intercept and regenerate by an unintended receiver. While the theory of m-sequences cannot be discussed in detail here, two circuits for their generation will be given and some of their properties listed. Figure 5 illustrates two feedback shift-register configurations for the generation of m- sequences. Each box represents a storage location for a binary digit, labeled with a D for delay (b) FIGURE 5: Two configurations of m-sequence generators: (a) high-speed linear feedback shift-register generator; (b) low-speed linear feedback shift-register generator [1]. book Mobk087 August 3, 2007 13:15 12 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION TABLE 3: Some Generator Polynomial Coefficients in Octal Format for m-Sequences; m = 2 r −1. DEGREE, NO. OF OCTAL REPRESENTATION OF THE r PRIMITIVE GENERATOR POLYNOMIAL POLYNOMIALS (g 0 ON THE RIGHT; g r ON THE LEFT) 21 [ 7 ] ∗ 32 [ 13 ] ∗ 42 [ 23 ] ∗ 56 [ 45 ] ∗ , [ 75 ] , [ 67 ] 66 [ 103 ] ∗ , [ 147 ] , [ 155 ] 718 [ 211 ] ∗ , [ 217 ] , [ 235 ] , [ 367 ] , [ 277 ] , [ 325 ] , [ 203 ] ∗ , [ 313 ] , [ 345 ] 816 [ 435 ] , [ 551 ] , [ 747 ] , [ 453 ] , [ 545 ] , [ 537 ] , [ 703 ] , [ 543 ] 948 [ 1021 ] ∗ , [ 2231 ] , [ 1461 ] , [ 1423 ] , [ 1055 ] , [ 1167 ] , [ 1541 ][ 1333 ] , [ 1605 ] 10 60 [ 2011 ] ∗ , [ 2415 ] , [ 3771 ] , [ 2157 ] , [ 3515 ] , [ 2773 ] , [ 2033 ] , [ 2443 ] , [ 2461 ] 11 176 [ 4005 ] ∗ , [ 4445 ] , [ 4215 ] , [ 4055 ] , [ 6015 ] , [ 7413 ] , [ 4143 ] , [ 4563 ] , [ 4053 ] ∗ Feedback connections from one intermediate delay. by T c s, and the summing circles represent modulo-2 addition. The connection circles, shown with a label g i in each case, are either closed or open depending on a generator polynomial g r g r −1 g 0 (1 signifies closed or a connection and 0 signifies open or no connection), where the g i s are coefficients of a primitive polynomial. Table 3 gives an abbreviated list of primitive polynomials to degree 11 (first column) with the total number of that degree given in the second column. The asterisks in Table 3, third column, denote feedback connections requiring only one adder. There are extensive tables of primitive polynomial coefficients to much higher degree [1]. book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 13 In Table 3, the primitive polynomial coefficients are given in octal format. For example, taking the first entry of the degree 10 listing, we have [ 2011 ] 8 ⇔ [ 010000001001 ] 2 ⇔ D 10 + D 3 +1. (4.1) All we want are the binary coefficients so that we know if a given connection is present or not in the shift-register circuits of Fig. 5. The particular 1s and 0s occupying the shift register stages after a clock pulse occurs are called states. Example2. An m-sequence of degree 3 is desired. Give the generator polynomial, the number of shift register stages, and the connections for the configurations of Fig. 5(a) and 5(b). Solution: From Table 3, the generator octal and binary representations and generator polyno- mial are [ 13 ] 8 = [ 001011 ] 2 ⇔ D 3 + D +1 = r−1  i=0 g i D i . The two generic forms of the sequence generators shown in Fig. 5 are specialized to this example and are shown in Fig. 6. Both generic forms have three delays in this example. Note that an initial load of 001 is assumed for the shift register of (a); subsequent states may then be found. The following properties apply to m-sequences: 1. An m-sequence contains one more 1 than 0. 2. The modulo-2 sum of an m-sequence and any phase shift of the same m-sequence is another phase of the same m-sequence (a phase of the sequence is a cyclic shift). 3. If a window of width r is slid along an m-sequence for N shifts, each r-tuple except the all-zeros r-tuple will appear exactly once. FIGURE 6: The two m-sequence shift-register configurations for Example 2. book Mobk087 August 3, 2007 13:15 14 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 4. Define a run as a subsequence of identical symbols within the m-sequence. Then, for any m-sequence, there are r One run of ones of length r. r One run of zeros of length r–1. r One run of ones and one run of zeros of length r–2. r Two runs of ones and two runs of zeros of length r–3. r Four runs of ones and four runs of zeros of length r–4. r r 2 r−3 runs of ones and 2 r–3 runs of zeros of length 1. 5. The autocorrelation function of a repeated m-sequence is periodic with period T 0 = NT c and is given by (0s replaced by −1values) R c ( τ ) = 1 T 0  T 0 x ( t ) x ( t +τ ) dt =     1 − | τ | T c   1 + 1 N  − 1 N , | τ | ≤ T c − 1 N , T c < | τ | ≤ N−1 2 T c , (4.2) where the integration is over any period, T 0 = NT c . 6. The Fourier transform of the autocorrelation function of an m-sequence, which gives the power spectral density, is given by S c ( f ) = ∞  m=−∞ P m δ ( f − mf 0 ) , f 0 = 1/NT c , (4.3) where P m =   ( N +1 ) /N 2  sinc 2 ( m/N ) , m = 0, sinc ( x ) = ( sin πx ) / ( π x ) 1/N 2 , m = 0. The autocorrelation function and power spectral density of a 15-chip m-sequence are shown in Fig. 7. Example 3. Consider the 2 5 −1 = 31-chip m-sequence: b =1010111011000111110011010010000. We see that it has 16 1s and 15 0s (property 1). The chip-by-chip modulo-2 sum of b and Db is computed as b = 1010111011000111110011010010000 D b = 0101011101100011111001101001000 b + Db = 1111100110100100001010111011000 . descriptions of two generic spread spectrum modulation techniques—QPSK spreading and frequency-hop spread spectrum. book Mobk087 August 3, 20 07 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 11 4 SPREADING. 0.5log 2 M ≈ 2 m  1 +cos ( π/M ) 2cos ( π/M ) ×Q  2m  1 −cos  π M  E b N 0  m = log 2 M, M > 2 11.2dB, M = 2 12. 9dB, M = 4 16.8dB, M = 8 21 .4dB, M = 16 26 .3dB, M = 32 M-ary phase- 0.5log 2 M ≈. hopefully. 3 TYPES OF SPREAD SPECTRUM MODULATION The two mostcommon types ofspread spectrum modulationare direct-sequence andfrequency- hop spread spectrum (FHSS). A binary direct-sequence spread spectrum